How to rig a cartoon-style hose limb? (cartoon arm)
I think you could test at least 3 solutions, the classic rigging with bendy bones playing with Easing In and Out, the Stretch To constraint solution, will allow you to stretch the arm, and the Spline IK constraint solution, which is a bit more complicated but will deform more nicely.

1) CLASSIC RIGGING


Create a chain of bones: upper arm, lower arm, hand, etc. Flex the elbow a bit so that Blender will know how it will have to bend.
Create a controller bone at the wrist position, deparent it and unclick the Deform option in the Properties > Bone panel.
Parent the hand to this controller with ctrl P > Make Parent > Keep Offset.
Create a Pole Target bone from the elbow, deparent it and move it a bit backward, unclick the Deform option in the Properties > Bone panel.
In the Properties > Data > Display panel, choose B-Bone instead of Octahedral. Give your upper and lower arms several Segments (for example 6). You'll see later that the Easing options will allow you to bend the arm in a rounder way. Maybe you'll have to correct the rotation of some bones with ctrl R so that they all follow the same axis.
In Pose mode, give your lower arm bone an IK Constraint, choose the controller as the Target, choose the Pole Target, and choose 2 for Chain. You can also give it a Copy Rotation constraint with the hand as the target, so that when you rotate the hand, the lower arm will follow.
Parent your mesh to the armature.
Don't forget to put the Armature modifier of your mesh above the Subdivision Surface modifier otherwise your mesh could be weirdly deformed (maybe it's the problem in your screenshot). Also, click Preserve Volume in the Armature modifier.
Now you can control the arm with the controller.
Sometimes it will go crazy because the Pole Target is not enough to show how to rotate, in that case select your arm, go in Pose mode, then in Properties > Bone > Inverse Kinematics, limit the rotation on the axis you want.




2) STRETCH TO CONSTRAINT

If you want your arm to be more free and elastic you could also try the Stretch To constraint:


Create a chain of bones with shoulder / upper arm / elbow / lower arm / hand / hand controller.
In the Properties > Data > Display panel, choose B-Bone instead of Octahedral. Give your upper and lower arms several Segments (for example 6). You'll see later that the Easing options will allow you to bend the arm in a rounder way. Maybe you'll have to correct the rotation of some bones with ctrl R so that they all follow the same axis.
The hand controller needs to be deparented, the hand is parented to the controller, the elbow must be deparented, the shoulder is parented to the torso of course. For the controller, unclick the Deform option in the Properties > Bone panel.
In Pose mode, give the upper arm and lower arm a Stretch To constraint.
In the lower arm constraint, select the hand as the Target. In the Properties > Bone > Bendy Bones panel, click the Use Custom Handle References and choose In > elbow and Out > hand.
In the upper arm constraint, select the elbow as the Target. In the Properties > Bone > Bendy Bones panel, click the Use Custom Handle References and choose In > shoulder and Out > elbow.
To test the arm, move the hand controller and the elbow, there are independent.
Parent you mesh to your armature.
Don't forget to put the Armature modifier of your mesh above the Subdivision Surface modifier otherwise your mesh could be weirdly deformed (maybe it's the problem in your screenshot). Also, click Preserve Volume in the Armature modifier.




3) SPLINE IK CONSTRAINT

Third method is Spline IK, a bit more difficult to set but very flexible and will probably preserve the volume better:


Create an armature as long as the arm, subdivide it for example 7 times, with one bone for the hand, deparent this hand bone
Create a curve as long as the arm (until the wrist), subdivide it once (so it has now 3 vertices), align it to the bones of the armature. In Edit mode make sure its direction is good, if not press W > Switch Direction.
Switch the armature to Pose mode, give the 6th bone a Spline IK constraint with the curve as Target and a Chain Length of 6 if you have chosen 6 bones for the arm. Now your curve controls your bones.
Choose the B-Bone display type for your armature, and segment your arm bones, for example 3 times.
Create a second armature, with one bone for each curve vertex.
Switch this armature to Pose mode, select one bone, shift select the curve, go in Edit mode, select the corresponding vertex, press ctrlH > Hooked To Selected Object Bone. Do it for the 3 vertices. Now the 2nd armature controls the vertices position. It allows you to come back to a rest position whenever you want.
Parent the 7th bone (hand) of the first armature to the 3rd bone of the second armature with a Child Of constraint so that this 3rd bone control the hand position and rotation. In the constraint, press the Set Inverse button so that the bone stays at its place.
Now parent the mesh to the first armature With Automatic Weight. Put the Armature modifier above the Subsurf modifier, enable the Preserve Volume option.